Shuffle approach to wreath Pieri operators

TL;DR

This paper explores a novel approach to studying wreath Macdonald polynomials and Pieri rules using quantum toroidal algebra, establishing new theoretical relationships and outlining computational methods.

Contribution

It introduces a new framework connecting wreath Macdonald polynomials with quantum toroidal algebra, extending classical pairing concepts to the wreath setting.

Findings

Established a relationship between dual wreath Macdonald polynomials and quantum toroidal algebra.

Outlined a method to compute norm formulas for wreath Macdonald polynomials.

Proposed a shuffle approach to Pieri operators in the wreath context.

Abstract

We describe a way to study and compute Pieri rules for wreath Macdonald polynomials using the quantum toroidal algebra. The Macdonald pairing can be naturally generalized to the wreath setting, but the wreath Macdonald polynomials are no longer collinear with their duals. We establish the relationship between these dual polynomials and the quantum toroidal algebra, and we outline a way to compute norm formulas. None of the aforementioned formulas are successfully computed in this paper.